In connection with the definition of the 3-body twist-3 DA (see, for instance, [1]}, [2]})
this relation between \(A_{\mu }\)  and \(G_{\mu \nu }\)  allows to derive the 3-body projector, \(q\bar{q}g\rightarrow \pi \) ,
to be used in perturbative calculations involving the \(q\bar{q}g\)  Fock component [3]}.
For an outgoing pion this projector reads
\({\cal P}_{3,l m}^{P(ab)\rho ,r}\,=\,\frac{-i}{g}\,\frac{f_{3\pi }(\mu ^2_F)}{2 \sqrt{2 N_C}}\,{\cal C}^{ab}_P \, \frac{\left(t^r\right)_{l m}}{C_F\sqrt{N_C}}\,\frac{\gamma _5}{\sqrt{2}} \, \sigma _{\mu \nu } q^{\prime }{}^{\mu } g_\perp ^{\nu \rho }\,\frac{\phi _{3\pi }(\tau _a,\tau _b,\tau _g, \mu _F)}{\tau _g}\,.\) 
